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This paper addresses the preservation of numerous essential properties of a residuated lattice structure in extended algebras for partial fuzzy set theory and partial fuzzy logics. The preservation includes the residuated lattice axioms, the identities narrowing the classes of the residuated lattices, and some well-known additional properties. In this paper, we consider nine algebras for partial fuzzy logics which incorporate handling undefined values in a bit different way. In particular, we consider the Bochvar, the Bochvar external, the Sobociński, the Kleene, the McCarthy, the Nelson, and the Łukasiewicz algebras, and two recently developed ones, namely the Lower estimation and the Dragonfly algebras. We summarize the obtained results in a comprehensible form which allows readers to easily check the information for the preserved and non-preserved properties in a certain partial algebraic structure. The resulting shape of the contribution is a sort of "atlas book" that aims at providing researchers with a comfortable and comprehensible form of an overview of the (non)preservation of fundamental properties of residuated structures. [ABSTRACT FROM AUTHOR]

The combination of fuzzy set theory and rough set theory has been discussed in a lot of research papers over the years. In this paper, we examine one such combination, namely the notion of rough L -fuzzy sets. We provide a representation theorem that determines when a pair of L -fuzzy sets is a rough L -fuzzy set, and we establish a connection between the lattice of rough fuzzy sets and the lattice of rough relations. Furthermore, we investigate the properties of the lattice of rough L -fuzzy sets and characterize the case when a three-valued Łukasiewicz-algebra can be defined on it. [ABSTRACT FROM AUTHOR]

We first make a great improvement on the early definition of a reducible element of a covering approximation space and formulate the definition of a reduction of the covering. Then the crucial links between the two concepts are established and a method for obtaining an optimal reduction of the covering is proposed. More importantly, we introduce the representation matrix for a finite covering approximation space and define a new type of operation on matrices. To obtain a reduction of the covering, we need only to deal with the representation matrix, so that the reduction of the covering can be executed on computers. The concept of the product of two covering approximation spaces is defined in this paper and we explore the connections between the reductions of the covering of the product space and those of the coverings of the factor spaces. Furthermore, we deal with the connections from the standpoint of algebra. A new type of matrix product is defined and the property of the product is explored. Drawing on the product, we can research the reduction of the product space. [ABSTRACT FROM AUTHOR]

In recent works, we have shown how to construct an information algebra of coherent sets of gambles, considering firstly a particular model to represent questions, called the multivariate model , and then generalizing it. Here we further extend the construction made to the highest level of generality, setting up an associated information algebra of coherent lower previsions , analyzing the connection of both the information algebras constructed with an instance of set algebras and, finally, establishing and inspecting a version of the marginal problem in this framework. Set algebras are particularly important information algebras since they are their prototypical structures. They also represent the algebraic counterparts of classical propositional logic. As a consequence, this paper details as well how propositional logic is naturally embedded into the theory of imprecise probabilities. [ABSTRACT FROM AUTHOR]

In this paper a number of implicative topological algebras have been developed which are based on the implicative quasi-Boolean algebra with operator (IqBaO) [10]. Their independence is established by several examples. Logics corresponding to these algebras are presented. Two new pairs of lower-upper approximations of a set have been introduced in order to develop the notion of duality with respect to the quasi-complementation. Set theoretic rough set models of some of the algebras are constructed using these lower-upper approximations and quasi-complementation. [ABSTRACT FROM AUTHOR]

In this paper we create a bridge between desirability and information algebras: we show how coherent sets of gambles, as well as coherent lower previsions, induce such structures. This allows us to enforce the view of such imprecise-probability objects as algebraic and logical structures; moreover, it enforces the interpretation of probability as information, and gives tools to manipulate them as such. [ABSTRACT FROM AUTHOR]

The paper introduces the notion of state for involutive bisemilattices, a variety which plays the role of algebraic counterpart of weak Kleene logics and whose elements are represented as Płonka sums of Boolean algebras. We investigate the relations between states over an involutive bisemilattice and probability measures over the (Boolean) algebras in the Płonka sum representation and, the direct limit of these algebras. Moreover, we study the metric completion of involutive bisemilattices, as pseudometric spaces, and the topology induced by the pseudometric. [ABSTRACT FROM AUTHOR]